Define a sequence recursively by $F_0=0$, $F_1=1$, and $F_n=$ the remainder when $F_{n-1}+2F_{n-2}$ is divided by $3$, for all $n\ge 2$. What is $F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}+F_{2025}+F_{2026}+F_{2027}+F_{2028}+F_{2029}$?